# Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup:

1. I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?

2. Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?

3. The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.

• I haven't got my copy of Weibel's book to hand, but IIRC there is something there about Hochschild cohomology for U(frak g) - perhaps that might help with 1. – Yemon Choi Jul 23 '18 at 23:33

The article Deformation par quantification et rigidite des algebres enveloppantes by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$ strongly rigid, and show that then every formal associative deformation is equivalent to the trivial deformation. For semisimple Lie algebras over an algebraically closed field of characteristic zero one has $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=H^2(\mathfrak{g},S\mathfrak{g})$, which is zero by Whitehead's second Lemma for Lie algebra cohomology.
1. Means that as a $\mathbb C[[\hbar]]$ associative algebra $U_\hbar(\mathfrak g)$ is isomorphic to $U(\mathfrak g)[[\hbar]]$. Each element can be understood as a formal power series $\sum_{n=0}^\infty \hbar^n X_n$ where each $X_n\in U(\mathfrak g)$ and the product is given by the usual product formula of formal power series $$\sum_{n=0}^\infty \hbar^n X_n \cdot \sum_{n=0}^\infty \hbar^n Y_n=\sum_{n=0}^\infty \hbar^n \sum_{j=0}^n X_j\cdot Y_{n-j}$$
2. Means that $U_\hbar(\mathfrak g)$ is a $\mathbb C[[\hbar]]$ Hopf algebra such that $U_\hbar(\mathfrak g)/\hbar U_\hbar(\mathfrak g)$ is again a Hopf algebra isomorphic (as Hopf algebra) to $U(\mathfrak g)$ . To see this as a deformation problem in a cohomological manner you should look for what is called Gerstenhaber-Shack cohomology. You can have a look at the original papers on the subject: